NCERT Solutions for Class 11 Maths Chapter 14: Mathematical Reasoning - Exercise 14.3

Exercise 14.3

1. For each of the following compound statements first identify the connecting words and then break it into component statements.

(i) All rational numbers are real and all real numbers are not complex.

Ans: Compound statements are those statements that are made up of two or more simpler or smaller statements. These smaller statements are complete in themselves and have their own independent meanings. 

Connecting words are those words which are used to connect two or more component statements of a compound statement. The connecting words are ‘AND’ and ‘OR’.

Consider the given statement, ‘All rational numbers are real and all real numbers are not complex’. 

The connecting word in the above compound statement is ‘AND’.

We will now determine the component statements.

Let the first component statement be P.

P: All rational numbers are real.

Let the second component statement be q.

q: All real numbers are not complex.

Therefore, the connecting word is ‘AND’.

The component statements for the given compound statement are,

P: All rational numbers are real.

q: All real numbers are not complex.

(ii) Square of an integer is positive or negative.

Ans: Compound statements are those statements that are made up of two or more simpler or smaller statements. These smaller statements are complete in themselves and have their own independent meanings. 

Connecting words are those words which are used to connect two or more component statements of a compound statement. The connecting words are ‘AND’ and ‘OR’.

Consider the given statement, ‘Square of an integer is positive or negative’. 

The connecting word in the above compound statement is ‘OR’.

We will now determine the component statements.

Let the first component statement be P.

P : Square of an integer is positive. 

Let the second component statement be q.

q: Square of an integer is negative.

Therefore, the connecting word is ‘AND’.

The component statements for the given compound statement are,

p:  Square of an integer is positive.

q: Square of an integer is negative.

(iii) The sand heats up quickly in the Sun and does not cool down fast at night.

Ans: Compound statements are those statements that are made up of two or more simpler or smaller statements. These smaller statements are complete in themselves and have their own independent meanings. 

Connecting words are those words which are used to connect two or more component statements of a compound statement. The connecting words are ‘AND’ and ‘OR’.

Consider the given statement, ‘The sand heats up quickly in the Sun and does not cool down fast at night’. 

The connecting word in the above compound statement is ‘AND’.

We will now determine the component statements.

Let the first component statement be P.

P: The sand heats up quickly in the Sun. 

Let the second component statement be q.

q: The sand does not cool down fast at night. 

Therefore, the connecting word is ‘AND’.

The component statements for the given compound statement are,

p: The sand heats up quickly in the Sun.

q: The sand does not cool down fast at night.

(iv) $x=2$ and $x=3$ are the roots of the equation $3{{x}^{2}}\left( n-x \right)\left( n-10 \right)=0$.

Ans: Compound statements are those statements that are made up of two or more simpler or smaller statements. These smaller statements are complete in themselves and have their own independent meanings. 

Connecting words are those words which are used to connect two or more component statements of a compound statement. The connecting words are ‘AND’ and ‘OR’.

Consider the given statement, ‘$x=2$ and $x=3$ are the roots of the equation $3{{x}^{2}}\left( n-x \right)\left( n-10 \right)=0$’. 

The connecting word in the above compound statement is ‘AND’.

We will now determine the component statements.

Let the first component statement be P.

$\text{p}$: $x=2$ is a root of the equation $3{{x}^{2}}\left( n-x \right)\left( n-10 \right)=0$. 

Let the second component statement be q.

$\text{q}$: $x=3$ is a root of the equation $3{{x}^{2}}\left( n-x \right)\left( n-10 \right)=0$.

Therefore, the connecting word is ‘AND’.

The component statements for the given compound statement are,

$\text{p}$: $x=2$ is a root of the equation $3{{x}^{2}}\left( n-x \right)\left( n-10 \right)=0$.

$\text{q}$: $x=3$ is a root of the equation $3{{x}^{2}}\left( n-x \right)\left( n-10 \right)=0$.

2. Identify the quantifier in the following statements and write the negation of the statements. 

(i) There exists a number which is equal to its square.

Ans: Quantifiers are the words that precede the nouns and are used to express the quantity of the objects.

Consider the given statement, ‘There exists a number which is equal to its square’.

The quantifier in the above statement is, ‘There exists’. 

The negation of a statement means to negate the statement. In other words, it is writing the opposite of the given statement. To write the negation of any statement, we usually add or remove the word ‘not’.

The given statement does not carry the word ‘not’. 

To write the negation of the given statement, we will add the word ‘not’ to it. 

Therefore, the negation of the given statement is, ‘There exists a number which is not equal to its square’.

(ii) For every real number $x$, $x$ is less than $x+1$.

Ans: Quantifiers are the words that precede the nouns and are used to express the quantity of the objects.

Consider the given statement, ‘For every real number $x$, $x$ is less than $x+1$’.

The quantifier in the above statement is, ‘For every’. 

The negation of a statement means to negate the statement. In other words, it is writing the opposite of the given statement. To write the negation of any statement, we usually add or remove the word ‘not’.

The given statement does not carry the word ‘not’. 

To write the negation of the given statement, we will add the word ‘not’ to it. 

Therefore, the negation of the given statement is, ‘For there exists a real number $x$ such that $x$ is not less than $x+1$’.

(iii) There exists a capital for every state in India.

Ans: Quantifiers are the words that precede the nouns and are used to express the quantity of the objects.

Consider the given statement, ‘There exists a capital for every state in India’.

The quantifier in the above statement is, ‘There exists’. 

The negation of a statement means to negate the statement. In other words, it is writing the opposite of the given statement. To write the negation of any statement, we usually add or remove the word ‘not’.

The given statement does not carry the word ‘not’. 

To write the negation of the given statement, we will add the word ‘not’ to it. 

Therefore, the negation of the given statement is, ‘There exists a state in India which does not have a capital’.

3. Check whether the following pair of statements is a negation of each other. Give reasons for the answer.

(i) $x+y=y+x$ is true for every real number $x$ and $y$.

(ii) There exists a real number $x$ and $y$ for which $x+y=y+x$. 

Ans: The negation of a statement means to negate the statement. In other words, it is writing the opposite of the given statement. Negation reverses the meaning of the statement. If a statement is false then its negation is true and vice versa. To write the negation of any statement, we usually add or remove the word ‘not’. 

Consider the first statement. 

The given statement is that, ‘$x+y=y+x$ is true for every real number $x$ and $y$’.

The given statement carries the word ‘not’. 

To write the negation of the given statement, we will add the word ‘not’ to it. 

Thus, the negation of the given statement is, ‘$x+y=y+x$ is not true for every real numbers $x$ and $y$’.

Now this statement is not similar to the second statement.

Therefore, the given pair of statements are not the negations of each other.

4. State whether the ‘Or’ used in the following statements is exclusive ‘or’ inclusive. Give reasons for your answer.

(i) Sun rises or Moon sets. 

Ans: Exclusive events are those events which are independent of each other and cannot occur simultaneously. On the other hand, inclusive events are those events which can occur simultaneously. 

Consider the given statement, ‘Sun rises or Moon sets’.

The ‘or’ in the above statement is exclusive. 

This is because the Sun cannot rise and the Moon cannot set together.

(ii) To apply for a driving license, you should have a ration card or a passport. 

Ans: Exclusive events are those events which are independent of each other and cannot occur simultaneously. On the other hand, inclusive events are those events which can occur simultaneously. 

Consider the given statement, ‘To apply for a driving license, you should have a ration card or a passport’.

The ‘or’ in the above statement is inclusive. 

This is because a person can have both a passport and ration card to apply for the driver’s license. 

(iii) All integers are positive or negative.

Ans: Exclusive events are those events which are independent of each other and cannot occur simultaneously. On the other hand, inclusive events are those events which can occur simultaneously. 

Consider the given statement, ‘All integers are positive or negative’.

The ‘or’ in the above statement is exclusive. 

This is because all integers cannot be both positive and negative. 

NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning

Opting for the NCERT solutions for Ex 14.3 Class 11 Maths is considered as the best option for the CBSE students when it comes to exam preparation. This chapter consists of many exercises. Out of which we have provided the Exercise 14.3 Class 11 Maths NCERT solutions on this page in PDF format. You can download this solution as per your convenience or you can study it directly from our website/ app online.

Vedantu in-house subject matter experts have solved the problems/ questions from the exercise with the utmost care and by following all the guidelines by CBSE. Class 11 students who are thorough with all the concepts from the Subject Mathematical Reasoning textbook and quite well-versed with all the problems from the exercises given in it, then any student can easily score the highest possible marks in the final exam. With the help of this Class 11 Maths Chapter 14 Exercise 14.3 solutions, students can easily understand the pattern of questions that can be asked in the exam from this chapter and also learn the marks weightage of the chapter. So that they can prepare themselves accordingly for the final exam.

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